cms – SEO Impact of two different systems of shopping cart under the same domain?

I had an old shopping cart website, url is something like www.example.com/shopsys/.
Later I have a new one using Opencart, still using the same domain name, but the landing page is at the root, i.e. www.example.com/.

The sold items are somehow similar, but many of the old one are not migrated to the new one, and the new one has some new additions. The bottom line is, the type of goods sold by the company hasn’t changed across these two sites.

The old shopping cart is still online, my boss said she wanted to preserve the SEO/page rank this old website has. Therefore, on every page of the old shopping cart, there is a huge banner reminding the visitor that he is in the legacy website, and clicking the banner will lead him to the landing page of the new website.

The problem is that I observed in Google Search console that most (about 60-70%) search impressions are from the old shopping cart.

For both the old and new shopping cart web, I have submitted sitemaps for both.

Should I put the old shopping cart offline, or should I just bring down the sitemap of the old shopping cart?

And any other suggestions? I am feeling that I might have been making some obvious SEO mistakes.

Thanks in advance!